In this context, the phrase "hour 11 of task X" means that you are in the 11th hour of doing the task. You've been doing it for somewhere between 10 and 11 hours (since the first hour goes from hour 0 to hour 1 - thanks Mathieu). The exact amount of time is not actually important to the author's point in the quote above, only that it's been a very long work day (especially at an unpleasant task).
If we instead say you performed an "11-hour task X" then we're saying that the task will be complete after 11 hours. Compare this to the "hour 11" version, which does not give any information about how long the task will have taken in total to complete (or whether it will complete at all!). The "11-hour" version also doesn't specify how much time we have already spent (only the total required) so we don't know if the task is already complete, in progress, or has not even started.
The key distinction here is that it's not a "time period" as the OP asked, but a "moment". The author chose to use this wording because he's talking about examining all of the moments in a person's life, and comparing lives which contain more pleasant (yoga) ones vs. lives that contain more unpleasant (fryer) ones. He says that if a randomly chosen moment from your life is more likely to be pleasant than unpleasant, then you are more likely to appreciate whatever the rest of the article is talking about.